Rotate (generated) UVs of environment texture without mapping node
This is a node setup for a z-axis rotation.  The other 2 axis' are just as easy but adding all three axis' into one formula gets very complicated.

To rotate the image, adjust the Value node that is located at the top-left in the "Deg 2 Rad" frame.

Note: The math nodes all have two "value" fields but only the upper field is used for trigonometry so just ignore any values in the lower fields.